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Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

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Abstract

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Marzieh Dehghani-Madiseh & Mehdi Dehghan

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  1. Marzieh Dehghani-Madiseh
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  2. Mehdi Dehghan
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Correspondence to Mehdi Dehghan.

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Dehghani-Madiseh, M., Dehghan, M. Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p). Numer Algor 73, 245–279 (2016). https://doi.org/10.1007/s11075-015-0094-3

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  • DOI: https://doi.org/10.1007/s11075-015-0094-3

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